Rep. Jim Jordan (R-Ohio), the chairman of the conservative House Freedom Caucus, said that there's a "good chance" the majority in his party will oppose the bank. Others within conservative ranks say they are about 30 votes from that goal.

What's unclear is whether the majority of the House GOP conference will fall in line with the most conservative members in opposing the bank even if its reauthorization is attached to a federal highway transportation funding bill that lawmakers must pass before July 30.

The Senate, where there is much more bipartisan support for the bank, is expected to attach an Ex-Im amendment to the highway bill in the coming weeks.

Jordan said that lawmakers in the House are not whipping votes. But conservatives working against the bank say privately that they have between 90 and 100 members on their side.

They would need to reach 124 in order to have the "majority of the majority" in the House GOP conference oppose the bank. Republican leaders frequently follow an informal rule where they don't vote on legislation if a majority of their party colleagues opposes it.

Rep. Ken Buck (R-Colo.), who is president of the House Republican freshmen, opposes the bank. Heritage Action, a conservative group that is also opposed, believe about 10 freshmen, including Buck, hold that position.

The business community and supporters of the bank assert that they still have a grip on the majority of House Republicans.

Three sources working on reauthorization said that certain reforms to the bank could provide cover for conservatives who are looking for political shelter.

One such source said that conservatives would consent to a stand-alone floor vote — which they have not done — if they were certain they had the votes to sink the bank.

"If opponents are so sure they have momentum, they need to call a vote," said the source.

Another source who is working to reauthorize the bank estimated that there were probably "maximum 40 percent of the House Republicans who oppose the bank."

Such an outcome would be about on-par with the 2012 vote — the last time House lawmakers voted on a stand-alone bill — when the legislation passed 330 to 93, with 147 House Republicans supporting it.

In June 2014, 41 House Republicans wrote to Boehner urging him to reauthorize the bank.

Still, Tea Partyers argue that they have momentum on their side. The expiration of the bank's charter prevents it from authorizing new financing projects, though it can still make good on its previous commitments.

The business community believes that Speaker John Boehner (R-Ohio) would not let the Tea Party kill the bank.

General Electric (GE), one of the bank's top beneficiaries, has a subsidiary plant near Boehner's district.

"If they prevail, it's the end of Boehner and it's the Tea Party running the House. If a Congressman from Ohio can't reauthorize Ex-Im then he might as well hand over the keys right then and there to the Tea Party," said the second source.

Boehner has said that he would allow for an open amendment process should the Senate attach a reauthorization amendment to the infrastructure spending bill.

Boehner's relationships with House conservatives remain tense following last month's trade policy debate, after which Republican leadership sought to demote several Tea Partyers from their posts for voting against the party line.

Buck was among those who ultimately ended up keeping his post, despite leadership pressure.

"We thought is was wrong for any punishment to take place [following trade] when people were simply voting their conscience on a policy issue," said Jordan.

"The best thing to happen, though, in my judgement is that Ex-Im never comes up and we get an orderly wind-down... [But] the Speaker has been clear where he's at. He wants to let this play out."